R 324 Philips Res. Rep. 12, 333-350, 1957

RESIDUAL' ENTROPY OF ICE,. .AND RELATED COMBINATORIAL PROBLEMS

hy J. L. MEIJERING 536.75: 549.511.1

SummaryThe theoretical residual entropy of disordered ice is about 5% higherthan Pauling's value R In (3/2). For the corresponding quadraticlattice the difference is 6%. Here a persistency effect is found, whichin Ag2H310o must give rise to unequal a-priori probabilities of thedipole orientations. The entropy computed for the graphite-typehoney-comh lattice is 20% higher than that found by Lipscomh.This difference is related to a boundary effect. It is shownthat Chang'sexpression for the entropy ()f diatomic molecules in a lattice can bederived directly with Pauling's method.

RésuméL'entropie rësiduelle théorique de la glace dësordonnëe est d'environ5% plus grande que la valeur de Pauling R In (3/2). Pour Ie réseauquadratique analogue la diffërence est de 6%. Ici on trouve un effetde persistance, lequel doit donner lieu à des probabilités à priori dif-férentes des orientations des dipoles dans Ie cas de l'Ag2H3106•L'entropie déduite pour Ie réseau en nid d'abeilles graphitique estde 20% supérieure à celle trouvée par Lipscomb. Cette différence estliée à un effet de bord. On montre que l'expression de Chang pourl'entropie de molécules diatomiques dans un réseau peut être dërivëedirectement avec la méthode de Pauling.

ZnsammenfassungDie theoretische Nullpunktsentropie von ungeordnetem Eis ist unge-fähr 5% höher als Pauling's Wert R In (3/2). Für das entsprechendequadratische Gitter ist der Unterschied 6%. Hier wird einen Persi-stenzeffekt gefunden, der in Ag2H3JOO zu ungleichen A-priori-Wahr-scheinlichkeiten der Dipolrichtungen führen muB. Die für das gra-phitartige Houigwabengitter abgeleitete Entropie ist 20% höherals die von Lipscomh gefundene. DieserUnterschied hängt mit einemRandeffekt zusammen. Es wird gezeigt, daB Chang's Ausdruck fürdie Entropie von zwei-atomigen Molekülen in einem Gitter unmittel-bar mit Pauling's Methode abgeleitet werden kann.

1. IntroductionIn ordinary ice the oxygen atoms lie at the Zn and S sites of a wurtzite

lattice. The corresponding cubic modification, where they form a diamondlattice, has heen estahlished more recently 1)2).

Each oxygen is linked to its four 0 neighbours hy "hydrogen honds".Bernal and Fowler 3) assumed that the hydrogens were not symmetricallyplaced hetween the oxygens on the O-H-O hond, hut nearer to one of them,in such a way that each oxygen has two "near" hydrogens. Pauling 4)showed that the numher of possible configurations corresponds closely tothe residual entropy found experimentally for ice.

334 J. L. MEIJERING

For the sake of simplicity we shall speak of "A-bonds" and "B-bonds",depending on whether the H is nearer to an oxygen at a Zn or an S siteof the wurtzite (or zinc-blende] lattice. The residual entropy per mole ofice ~sthen given by the number of possible configurations ofA- and B-bondsin the Iattice.. if two A-bonds and two B-bonds meet at each of the Nlattice points *). Without this restrietion there would he 22N configurationsof the 2N bonds. The probability that two A- and two B-bonds meet inone point is then i.

Pauling simply takcs the probability that this condition is fulfilled inall N points to be equal to (3J8)N, yielding (3/2)N configurations and thusthe entropy S= RIn (3/2).

Pauling's secondmethod goes as follows. There 'are six ways of attachingtwo A- ~nd two B-bonds to a lattice point (six possible directions for theH20 dipole). The 6N possible combinations are multiplied by (lJ2)2N toaccount for the necessity that the designation (A or B) conferred on a bondmust be the same for both lattice points involved. Again S = R In (3/2)is found.The two methods need not, per se, give the same result. If it is stipulated,

for instance, that three A-bonds and one B-bond should meet in each latticepoint **), the first method yields R~(3/2) In 3 -In 4~, that is, 40% lessthan 2R In (5/4) found by the second method. The discrepancy is probablyconnected with the non-equivalence of A- and B-bonds in this case. In amodel withfour different bonds (say: C,D,E and F) meeting in each point,the two methods yield the same result, again S = R In (3/2).

The fact that this CDEF-model and the "A2B2" (ice) model should havethe same entropy according to Pauling's methods is remarkable, becauseit can he seen that in reality this is not so:

Consider a configuration of the ice model. The A-bonds (and also theB-bonds) form loops, either closed loops comprising an even number ofbonds, or loops beginning and ending at the surface of the crystal. We nowreplace the A-bonds in the loops by alternate C- and D-bond~; likewise theB-bonds by alternate E- and F-bonds. Each A-loop (e.g.) can obviouslybe ,replaced in two ways by a CoD-loop. Consequently 2M CDEF-configura-tions can be made from one and the same A2B2-config~ation, where Mis the number of loops in it. A conservative estimate of the average valueof M gives something of the order of 0·01 N. Therefore Pauling's methods(even if they do give concordant results) cannot be exact for at least oneof the two models considered. This is not surprising, since probabilities are

*) N is Avogadro's number, or, more generally, the (very great) number of l~tticepoints.

**) This model might possibly correspond to the physical case of ammonium fluoride,see 17).

RESIDUAL ENTROPY OF ICE, AND RELATED COMBINATORIAL PROBLEMS 335

assumed without justification to be independent of each other. However,the samé result S=RIn (3/2) was found in ä more elaborate way by Slaterin a paper 5) on KH2Pû4, the high-temperature equilibrium of which wouldbe quite analogous to the case of cubic ice. .

In the general KH2Pû4 case the entropy found by Slater (cf. his equa-tions (3.10) and (3.11)) can be written in the form

S = R ~-'-P+ In P+ - P_ In P_ - 4"n In pn. ++ 2 (P+ + 2Pn) In (P+ + 2Pn) + 2 (p_ + 2PII) In (p_ + 2Pn)~' (1)

Here P+ and P_ are the fractions of dipoles. pointing upwards and down-wards respectively along the tetragonal axis, and Pn = -!-Po is the fractionof dipoles pointing in one of the four directions at right angles to that axis.Exactly the same equation can be found by Pauling's second method, if

the 6 possible combinations of two A- and two B-bonds joining in one latticepoint are not equally numerous but are in the proportions P+ :P_: Pn: Pn: Pn :'pn.Slater's general result (1), and not merely for P+ = P_ = Pn = 1/6, isthus identical with that obtainable by Pauling's method. Indeed, thereappears to be no essential difference between their methods. In buildingup the crystal by layers, Slater discriminates between bonds going upwardsfrom one point - this giving rise to a correlation between their kinds(A, B) - and bonds coming up from below and just happening to meeteach other, which are thus uncorrelated in kind. Such a procedure cannotbe quite correct.

Slater's paper has apparently resulted in the acceptance of RIn (3/2) asthe exact theoretical value for the residual entropy of ice without order(cf. 6)7)8)9)10)). Recently, doubt has been expressed, however 20).In this paper it will be shown that the theoretical entropy is in fact

higher than Rin (3/2). But before going int"a the case ofice proper, we shallfirst consider the corresponding quadratic lattice. The treatment of thistwo-dimensional model is simpler, and the correct value for the entropycan be arrived at in closer approximation than for the three-dimensionalice lattice.

2. The quadratic pseudo-ice lattice; modified Pauling method

The number of configurations ofthis lattice (cf. fig. 1), with two A- andtwo B-bonds meeting in each lattice point, is again found to he (3/2)Naccording to Pauling's methods. Indeed, the geometry of the lattice playsno role in these methods, apart from the coordination number four.

It is rather easier to treat the duallattice, formed by the centres of thesquares of the lattice proper. They are geometrically identical, but in theduallattice each square in fig. 1 must have two A- and two B-sides, theentropy problem then being completely equivalent ..

336 J. L. lIIEIJERING

The number en of possible configurations of a single horizontal stripof n consecutive squares is 2.3n. The factor 2 can be omitted when n is ofthe order of iN and we are interested only in the logarithm. The numberof different configurations of the upper (or lower) edge .of the strip is 2n.To each of these corresponds a number of strip configurations, Ci. We have

2n

~ Ci = en = 3n.i=l

(2)

92317

Fig. I, Quadratic lattice.

The configuration number e2n of double strips with 2n squares (seefig. 2) is found by fitting together two single strips, the upper edge of onebeing identical with the lower edge of the other. Because of symmetry Ci

is the same on either side, so that

(3)

I I I I I ~92318

Fig. 2. Double strip. In the dual A2B2-lattice each square must have two A- and two B-sides.

It can be easily proved that this quadratic expression - for constant~Ci - is minimum when all ci-values are equal, that is to say when thesequence of bonds (in terms of "A" and "B") in an edge of a single strip israndom. This is not the case, however, because of the condition that eachsquare must have 2 A- and 2 B-edges. For instance, Ci is only n (instead ofthe mean value (3j2t) for an edge consisting of alternate A- and B-bonds.And for an edge consisting of A-bonds. only we find Ci = (1·618t *).

*) In the sequence of n parallel transverse bonds no successive A-bonds may,appear. Wecalculate thenumber ofwaysinarranging (n-2x) "B's" andx "AB's" (not also "BA's").Differentiation with respect to x then yields the maximum value, ei= [t(VS+l)]".

(5)

RESIDUAL ENTROPY OF ICE, AND RELATED COMBINATORIAL PROBLEb1S 337

Therefore(4)

By fitting two double strips together, etc. etc., a lattice can be built upwhich is .also effectively infinite in the vertical direction. With eachdoubling of the strip width a quadratic expression is obtained which isminimum if the 2n configurations of the middle edge are equally probable.In total for N squares (N~ n) we have

and thus S> R In (3/2).Something more quantitative can he derived by proceeding as follows.

Of the 18 possible configurations of a "double-square" it can easily beshown that 5 have an A-A upper edge, 5 a B-B, 4 an A-B and 4 aB-A.The probability that two arbitrary double-squares on top of each other havethe same long edge (so that they "fit" together) equals 2(5/18)2 + 2(4/18)2= 82/324, while 1/4 = 81/324 would be found if all 4 single squares weretreated as independent. By dividing the lattice into tN double-squareswith the same orientation an approximate value for the entropy can befound by assuming that the probability of fit is 82/324 for each long sideof the rectangles, independently of each other. For the short single sidesthe probability is taken simply to be t.We then find S= tR In (41/18).

The procedure itself is the same as Pauling's, but the unit is a double-square instead of a square. The larger this unit is chosen, the better the resultwill be. Table n shows -the values of S' = S/(R In 10) *) obtained withunits of up to 5X 4 squares.

The frequencies of occurrence of the different edges were systematicallyevaluated with the help of table I.For example, the sub-table Ib for tripleedges implies that if the upper side of a 3X1 rectangle is AAA ("homo-geneous"), 3 of the possible lower sides are homogeneous, 1 is "(heteroge-neous) symmetrical" (e.g. ABA) and 4 "asymmetrical" (e.g. AAB), etc.It should be noted that there are four different asymmetrical edges againsttwo of each of the other two sorts. Therefore half the numbers of homoge-neous, symmetrical and asymmetrical lower (or upper) edges become 8,5 and 14. Now, for rectangles composed of 3 X 2 squares, half these num-bers are easily found to be 8 X 3 + 5 X 1 + 14 X 2 = 57, 8 X 1 +5 X 2 + 14 X 1 = 32, and 8 X 4 + 5 X 2 + 14 X 4 = 98, respectively.III all there are 2 X 187 possible "3 X 2" rectangles. Of the short (double)edges of these rectangles 105/187 are homogeneous and 82/187 heteroge-neous. The numbers 105 and 82 are arrived at as follows. The hom.jhet.ratio is 5: 4 in "2 X 1" rectangles (see above, and sub-table la). It is

*) We use (log CN)/N, called S' for short, instead of kIn C = S to facilitate the numericalcomputations.

J. L. JlIEIJERING

TABLE I:' Used for deriving probabilities of different sorts of edges. Suh-tablesa, b, c and d apply to rectangles 2, 3,4 and 5 squares broad, respectively.ho = homogeneous; he = heterogeneous; sy = symmetrical; ,as = asym-metrical .

3 1 1 0 4 2 4 0 2 4

1 2 1 0 2 2 2 2 2 2

1 1 2 1 2 2 2 2 0 2

0 0 1 2 0 2 0 2 0 0

2· 1 1 0 4 1 3 1 2 3

1 1 1 1· 1 3 2 2 1 1

2 1 1 0 3 2 4 1 2 3,

3..

: 0 1 1 1 1 2 1 1 0

1 1 0 0 2 1 2 1 3 2>---

2 . 1 1 ·0 3 1 3 0 2 . 4,..

ho he

\,::ffiBa

" ho sy as

ho 3 1 4

1 2 2

2 1 4

sy

as

b

",

3 2 2 0 4 2

2 2 2 0 4 2

2 2 2 0 2 2

0 0 0 2 2 2

2 2 1 1 3 2

1 1 1 1 2 3

c

d

RESIDUAL ENTROPY OF ICE, AND RELATED"COMBINATORIAL PROBLEMS 339:

(5X 3+4 X2) : (5X?+4 X2) = 23 : 18 in "2 X2" rectangles and (23X3+18X2): (23X2+18X2) = 105: 82 in "2 X 3" rectangles. The modifiedPauling method with this unit yields cönsequently

, ,

S' = t pog (572 + 322 + 492 + 492)' ++ log (1052 -+.- 822)....,.. 3 Iog 1'87-log 2~= 0·18174.

When S' for "n X I" units is plotted versus l/n the points for n = 2 to5 lie within 10-5 on a straight line which is extrapolated to S' = 0·1820for l/n = O. This approximates to the result obtained with an infinitesingle strip as unit.

TABLE II

S' = S/(R In 10) of the quadratic A2B2-lattice calculated with differentunits of mXn squares '

1 0·17609m 2 0·17876 0·18Ö63t 3 0·17982 0·18174 0·18269

4 0·180~6 0·18244 0·18332 0·183885 0·18068 0·18290 '0·18377 0·18428

1 2 3 4n ___,..

In fig. 3 the S' -values 'for "m X n" units with m and n > 1 are plottedagainst 1/(m+n-l). The S'-values with m = 2 and m= 3 lie (within2.10-5 and 10-5, respectively) on straight lines, which are extrapolated toS' = 0·1852 and 0·1865 for n = 00. But for m=4 only the three highestpoints lie equally well on a straight line, leading to S' = 0,1872, and thepoints with m = 5' are ~seless for extrapolatory purposes.The values obtained for infinitely long strips with m = 1 to 4 (0'1820,

0,1852,0,1865, 0·1872fappearto ext~apolateto S' = 0'188forthe quadraticA

2B

2-lattice. ' . . .

3. Fowler-Rushbrooke method

The result of the double extrapolation in section 2 is rather uncertain.Therefore S' for the quadratic A2B2-lattice has been calculated also' inanother way. The method is essentially the same as that used by Fowlerand Rushbrooke 1~)for computing the entropy of iN diatomic moleculeswhich completely cover a lattice (see seëtion 9):'

Let Pn be the number of configurations of a strip 2' squares ~d~ au'd 'nsquares long, terminating in a homogenéous ene! (A-A ~orB-B), and Qn.

340 J. L; lIlEIJERING

thè same but with a heterogeneous end (A-B or B-A). We then have (cf.section 2, sub-table Ia)

P n+l = 3 P n + 2 Qn, 2Qn+l = 2 P n + 2 Qn. S

By putting Pn =PP~ and Qn = qp~ we obtain the equation

,u~- 5P2 + 2 = o.

(6)

(7)

w~o~----~a~,------~a~.2------~a~.3~~1j(m+n-l)_ 92319

Fig. 3. The S'-values of table II with m and n> 1 plotted versus lJ(m+n-l). The(interchangeable) values of m,n are indicated.

The configuration number for effectively infinite n is P~, where P2 is thelarger root of (7).

The equations for strips 3 and 4 squares wide and analogous to (6) canbe written by using the sub-tables 1b and le respectively. They reduce to

P~- 9,ui + 15p3 - 6 = 0and, eliminating a factor P4 + I,

,u~-16,u: + 65p: - 92p: + 48p4 - 8= o.

(8)

(9)

RESIDUAL ENTROPY OF ICE, AND RELATED COMBINATORIAL PROBLEMS 341

The values of #2' #a and #4 found from these equations were in perfectaccord (logarithms within 10-S) with those obtained less directly. If wecall e.g. C4,s the number of configurations of a 4 X 5 rectangle, the ratiosC4,2/C4,l' C4,a/C4,2' ... , C4,lO/C4,9 converge rapidly to #4' The ne?essary

. C-values were available as a result of the computations of section 2.In this way #s and #s were determined from the data up to Cs,s and CG,srespectively.

The values of IOg(#2!1tl)' log(#a/ #2)"" must converge to Sf of the lattice ..They are: 0·1819?, 0,18424, 0·18533, 0·18594 and 0·1863, the last number,log({ts/{ts)' being less accurate. This series extrapolates to 0·187, a valuemore reliable than the 0·188 found in section 2. Thus the entropy of thequadratic A2B2-lattice is found to be 0·187 R In 10, which is 6% higherthan the Pauling value.

4. Non-equivalence of possible "dipole" orientations' in the quadratic lattice

In one respect the quadratic lattice is less simple than the three-dimen-sional ice lattices proper. There, all six combinations of bonds meeting inone point (dipole orientations) will be equivalent, but in the quadraticlattice we have only four equivalent "dipole" orientations while the twoother combinations haye no "dipole moment". In those two - which wecall para groupings - the two vertical bonds are A-A and B-B respectively;in the dual lattice they are represented by squares having equal sidesparallel. Analysis of the possible configurations of the 2 X 2- and 3 X 3-square units show that the para squares appear with a frequency greaterthan 1/3. The central square of the latter unit is para in 35·8% of the 2604configurations. By the Fowler-Rushbrooke method (cf. section 3) thecentral row of squares of an infinitely long strip three squares wide is foundto show a para fraction of 36·2%. It seems rather safe to assume that thisfraction is at least 36% in the infinite lattice.

In a string of A-bonds in the quadratic lattice proper (not the dual one)three paths can be taken on arriving at each lattice point: left-hand turn,right-hand turn, and-straight on. The latter has a slight preference.

5. Silver-trihydrogen paraperiodate and ice

Ag2HaIOs has been found12) to be rather analogous to KH2P04. Theentropy associated with random arrangement of the hydrogens correspondsto the number of configurations of a simple cubic lattice with 3 A- and 3B-bOJids meeting in each lattice point. There are 20 configurations of asingle HaIOs group, namely 12 with a dipole in a [lOO]-direction and 8with a dipole in a [Ill]-direction *). The Pauling methods yield12) a residual

*) In reality the Ag2H3IOs lattice is rhombohedral, but topologically the H3IOa2- ionsmay be treated as lying on a cubic lattice.

342 J. L. lIIEIJERING

entropy óf RIn (5/2), but, as in section 2, it can be shown that the realvalue should be higher. Furthermore, in their equation on p.14I6 Stephen-son and Adams have implicitly assumed equal a-priori probabilities for all20 dipole positions. This is not justified. There will be a persistency effect,as in the quadratic case, and consequently the [lll]-dipoles will have asomewhat lower statisfical weight than the [IOO]-dipoles. From a generalstandpoint it is of interest to point out that a structure may' consist ofunits which acquire unequal a-priori probabilities by an entropistic coupling.

In the cases of hexagonal ánd cubic ice it can again be proved on thesame lines as in section.2 that the Pauling value RIn (3/2) is too low. Com-putation of the real value would be very laborious for these three-dimen-sional lattices. Some calculations (inter alia with the modified Paulingmethod, cf. section 2) lead to the conclusion that S is at least 2% higherthan RIn (3/2). But the difference will probably be less than that in thequadratic case, 6%. We now show why it is plausible thathexagonal iceshould have a smaller S than the quadratic A2B2-lattice ..Both lattices can be derived from the cubic lattice mentioned above.

By abolishing all vertical bonds, a series of parallel sheets of the quadraticlattice is obtained. But by abolishing a third of all bonds in another patternit is possible-to obtain the lattice sketched in fig. 4a" which is topologicallyidentical with hexagonal ice. The bonds to be cancelled now lie on linesrepresented by fig. 4b.

The configuration numbers of the two lattices are equal to those of acubic lattice with two A-, two B- and two C-bonds meeting in each point,additional conditions being that the C-bonds lie on parallel straight lines,

92320 a bFig. 4. a: Sheet of a lattice topologically identical with the wurtzite lattice. The parallelsheets are to he connected hy honds going up from the lattice points marked by a crossand down from those marked by a dot. Crosses ánd dots interchange in alternating sheets.By filling in honds lying on Iines.pictured in b (main direction perpendiëular to the sheets)a simple cubic lattice is obtained, .

M

RESIDUAL ENTROPY OF ICE, AND RELATED COMBINATORIAL PROBLEMS 343

o~ (for ice) on the fig. 4b-type lines making right angles everywhere. The. consideratiöns on persistency in direction (see above) make it highlyprobable that the entropy will be somewhat lower in the case of ice. Thisconclusion is supported by section' 6. It may he added that the differencein ~ between hexagonal and cubic ice will be insignificant.

6. Derivation of the residual entropies of ice and quadratic "pseudo-ice" by, a "building-up" method '

Suppose a cubic ice crystal be cut along a (lOO)-plane, through NK infig. 5, perpendicular to the paper. Let all bonds below the plane be desig-nated A- or B-type. The next layer of bonds is now built up and consistsof zigzags with [011] as main direction, inter alia ... NLKM ....

Fig. 5. Diamond lattice. G and H lie on either side of the plane through NLKM.

If GK an HK are both A-bonds, KL and KM must both be B-bonds.But if GKH is heterogeneous (AB or BA), we have a two-fold choice forLKM. As i of the bond pairs in cubic ice are heterogeneous, the numberof configurations of the new layer with n dipoles is 22n/3• At first sight thiswould appear to lead to an entropy of iR In 2 (S' = 0·2007) for one moleof ice. This is not correct, however. By making the two-fold choices atrandom, the new bond pairs (NLK, etc.) will be formed with a heteroge-neous frequency' less than f. If a large number of layers is built up and anattempt is made to render the entropy a maximum, the two-fold choicesmust not be completely random. By trying to attain too great an entropy,the situation is spoiled for later layers.

This should have two consequences for the surface properties of cubic ice, and similar.effects may be expected for normal ice.

(i) While in the crystal interior the dipole axis directions occur with equal frequency inthe six <100>-directions, at the (lOO)-surface they tend to favour a direction perpen-dicular to the surface.

(ii) There is a positive contrihution to the surface entropy.Of course these two effects might largely be overshadowed by energetic and vibrational-

entropy effects. .

344 J. L. lIrEIJERING

Suppose the layer is built from left to right. When NL has been desig-nated A or B, and GKH is a heterogencous bond pair, LKM is chosen tobe either AB or BA, not at random, but with a 2 - to - 1 statistical bias, inorder to make i of the new bond pairs (NLK, etc.) heterogeneous.

Here it is implicitly assumed that if, e.g., GKH is a homogeneous bond pair, so that thebond type of KL is fixed, the probability that NLK is heterogeneous will also be t.In a natural crystal, the correlation between bond pairs like GKH and NLK (in terms of"heterogeneous" and "homogeneous", not specifying A- or B-bonds!) will presumably berather weak. In any case the whole procedure is not exact, because in designating LKMas AB or BA it is not sufficient merely to consider the bond type of NL.

This building-up method leads to

thus S' = 0·1843.When applying the same method to the quadratic A2B2 ("pseudo-ice")

lattice, building up in the diagonal direction, thé persistency effect mustbe taken into consideration. We have

S = - ~-R (t In t + -i In -!) ; (10)

S= -Rx ~xlnx + (I-x) In (l-xH, (11)

where x, the heterogeneous bond-pair fraction, is at least 0,68, correspond-ing to a "para fraction" of at least 0·36 (cf. section 4). Thus S' should ex-ceed 0·1851. Expression (11) is maximum for x= 0,7035, with S' = 0·1857.The agreement with the computed value of 0·187 (cf. section 3) shows thatthe building-up method gives not too bad an approximation. Equation(10) can be expected to yield a slightly too low value for the residual entropyof ice itself, probably about 0·85 cal/degree against 0·805 resulting fromPauling's approximation.Although the difference probably does not exceed the uncertainties in

the experimental residual entropies of ice 13) and heavy ice 14), it is helpfulthat the theoretical value for the disordered Bernal-Fowler-Pauling modelis increased. This leaves more room for partial order of the dipoles, thedesirability of which ,was stressed inter alia by Gränicher 9), who arguedthat the experimental residual entropy might perhaps be 0·1 cal/degreetoo high. It appears difficult to believe that a virtually random arrangementof the dipoles could be in thermal equilibrium either at the lowest tempera-tures of specific-heat measurement or in the region where equilibrium wouldpractically "freeze in". However, there is a good deal of controversy on thedipole interaction energies and on the "freezing-in temperature" 6)15)10)9).In applying the "building-up" method to cubic ice' it has been tacitly

assumed that the fraction of heterogeneous bond pairs in the horizontalzigza~s (... GKH ... ? ••• NLKM ... ? etc., in fig. 5) amounts to °i, the

;-_--:"__,,=--------------------~---~ ..~-~~-

RESIDUAL ENTROPY OF ICE, AND RELATED COMBINATORIAL PROBLEMS 345

heterogeneous fraction of all hond pairs. This is not necessarily so; thecubic structure might actually he tetragonal as regards the hydrogen po-sitions. According to equation" (11), with x now standing for the heteroge-neous hond-pair fraction in horizontal zigzags, this should indeed he thecase, hecause the ent~opy will he increased hy x > 2/3. However, this wouldhe taking the "building-up" approximation too seriously: The questionwhether athermal models (that is to say, all micro-states having equalenergy) could show long-range order is interesting. But presumahly therewill he only short-range order in the models considered heré ..When ener-gies come into the picture, however, the formation of long-range order willhe influenced appreciahly hy entropistic ordering tendencies. And in the.case of KH2P04 Slater's treatment 6) may well he more incorrect than thefact that the resulting entropy for high temperatures is only 5% too lowwould indicate.

7. Models amenable to exact treatment

It is worth mentioning a group of models where hoth Pauling methods(cf. section 1) and the "building-up" method (cf. section 6) yield the sameresult and which, furthermore, can he proved to he exact.

Hit is stipulated that an even numher of A-honds (and thus also an evennumber of B-honds) meet in each point of the quadratic lattice, all threemethods mentioned are found to yield R In 2 for the entropy. This is shownto he correct as follows: Confer the designations "A" and "B" on all Nhorizontal honds. Then in each vertical line these designations are fixedas soon as the designation has heen chosen for one hond only.Exactly the same is found if it is stipulated that there must he odd num-

hers of A-honds and B-honds in each lattice point. Also, it is immaterialwhether the lattice is quadratic or three-dimensional with coordination 4,e.g. diamond. The sequence of hond types is random along each path com-posed of honds, provided the path does not interseet or touch itself.

Similar results are found for "even" and "odd" models with a highercoordination number z. The entropy is

s= (tz-1)Rln 2 (12)

and the same equation holds for "even-odd" models with odd coordinationnumher z. Thus, for instance, the number of configurations of a honey-comh lattice (cf. fig. 7) where either 3 A-honds or 1 A-hond and 2 B-hondsmeet in each point is 2!N, as can he shown by all four methods mentionedahove. This numher of configurations is the numher of ways in which un-connected closed loops (and loops heginning and ending in the latticebo~ndaries) can be arranged in the lattice,

346 J. L. MEIJERING

For this and the "even" square and triangular models, equation (12) ismost easily derived by painting the (tz - l)N elementary cells enclosedby bonds white or black, at random. Bonds separating white and blackcells are B-bonds, the other A-bonds.

8. Rundle's polar ice model and graphite

RundIe 7) has proposed for hexagonal ice a model differing from Pauling's ..It is polar, all bonds in the c-axis direction being, say, "B-type". In thepuckered basal layers two A-bonds and one B-bond meet in each latticepoint. Lipscomb 8) observed that the entropy problem associated herewithis identical with that of finding the number of resonance structures ingraphite. .

Another source of disorder in Rundle's model is associated with the fact that the anglebetween the bonds is smaller than the tetrahedral angle, and contributes iN In 2. Lips-comb found 0·95 cal/degree for the total residual entropy. According to our computations(see below) this entropy should even he 1·00 cal/degree, thus strengthening Lipscomb'sargument that Rundle's model has an appreciably larger residual entropy than the experi-mental one.

Gordon and Davison 16) have calculated the number of resonancestructures of several types of reticulate aromatic hydrocarbon. For mole-cules with a vicinal perimeter in the form of a regular hexagon (cf. fig. 6)this number is given by an expression derived by Woodger 16). Its logarithmcan be written as

where n is the number of small hexagons along one of the sides of them:olecule.

Lipscomb computed the .entropy for effectively infinite n and found 0·26

Fig. 6. Big hexagonal aromatic molecule (n = 4.in equation (13» having 232848 reso-nance structures 16). One double and two single bonds have to meet at each point, 24 C-Rbonds projecting from the perimeter. .

RESIDUAL ENTROPY OF ICE, AND RELATED COMBINATORIAL PROBLEMS 347

calJdegree. This agrees with the value R(! In3 - In 2) = 0·2599 calJdegreewhich is pasily found by transforming the summations in (13) into inte-grations.But the question is whether the result is pertinent. Gordon and Davison;s

results apply to hydrocarbons, and for their derivation it is essential thatall bonds projecting from the molecule perimeter are single. Thus the en-tropy R(! In 3 -In 2) applies to the big hexagon with two A- and oneB-bond joining in each lattice point, with the additional condition that allfree bonds at the perimeter must be A-type. Apparently Lipscomb took itfor granted that this condition has no effect, because the fraction of bondsconcerned vanishes for infinite n. Some doubt arises, however, in consider-ing non-hexagonal shap~s of the big molecule. For a large rhomb withvicinal angles 1200 and 600 the number of configurations 16) is so smallthat the entropy becomes zero when it is stipulated that the free honds atthe rhomb perimeter must be A-type.

Fig, 7. Honey-comb lat~ice (drawn) with its dual triangular lattice (dashed).

Therefore the entropy of the A2B-model without this additional conditionwas computed with the modified Pauling method. As in section 2, the duallattice was employed, i.e. a triangular lattice (cf. fig. 7), each elementarytriangle having one B- and two A-sides. Equilateral triangles comprising4, 9, and 16 small triangles were used. The corresponding values of S'plotted against the reciprocal of the unit area lie well on the upper straightline in fig. 8. The lower line refers to rhomb units consisting of two equilat-eral triangles. The lines fail to extrapolate to' the same value for infiniteunits. Presumably they should be curved towards each other. This issupported by the fact that, outside the field of fig. 8, the point representingthe single-triangle unit (nu = 1) lies markedly below the upper line andthe point repreaenting the smallest rhomb (nu = 2) lies markedly ahovethe lower line.

348 J. L. lI1EIJERING

But S' =0·068 is probably within 1% of the real value fortheA2B-model;it is 20% higher than Lipscomb's value (S' = 0·057). This pronouncedinfluence of the perimeter conditions is easiest discussed by consideringthe problem of covering a honey-comb lattice completely with diatomicmolecules. This is equivalent to the A2B-model (cf. section 9). Lipseemb'svalue stands for the entropy associated with filling the N sites of a bighexagon with iN molecules, taking care that no molecule is astride theperimeter and thus no marginal site remains unoccupied. The 20% greaterentropy derived above corresponds to the case when these "raw ends" 11)are not shunned.

0.077

0.073

V./-:

-:V/

1/

I---~ r-----

0.069

ts·0.065

0 0.1 0.2'12324

Fig. 8. Graph for computing S' = S/(R In 10) of the A2B honey-comb lattice. nu is thenumber of elementary triangles (cf. fig. 7) in the unit which is in the form of a rhombat the lower line and triangular at the upper line.

The latter method is preferable, because however large the energiesassociated with lattice faults may be, at finite temperatures they willalways be overcompensated by the higher entropy per mole when the crys-tal is sufficiently big. The perimeter effect in question cannot be inter-preted simply in terms of surface, edge, and corner entropies and energies(as in apparent from the fact that the entropy of a rhomb crystal wouldbe zero, see above), but it acts in a direction opposite to the normal ther-modynamic effects of lattice boundaries.

In the case of Rundle's ice model there is anyway no reason why thebonds projecting from the surface of a hexagonal prism should be of thetype opposite to that of the bonds in the c-axis direction.

Eindhoven, May 1957

RESIDUAL ENTROPY OF ICE, AND RELATED COMBINATORIAL PROBLEMS 349

9. Entropy of mixtures of single and double molecules

Fowler and Rushbrooke 11) were the first to compute the entropies ofathermal mixtures of single and double molecules on the square latticeand some others, although not on the honey-comb lattice. The entropyinvolved in filling a lattice, having a coordination number z, with doublemolecules only is clearly the same as that of the corresponding AZ--1B-model in our notation.

Chang 18) derived a closed expression for the entropy of tNx doubleand N(I-x) single molecules on N sites:

S=R~(x-tz) In z-tx In x-(I-x) In (I-x) + t(z-x) In (z-x)~. (14)

Several authors have since (see 19)) extended the calculations to triatomicmolecules, etc., and have, moreover, simplified Chang's derivation. It willnowbe shown that (14) can be'derived in a still easier and more direct wayby a simple extension of Pauling's first method (cf. section 1).

There are tNz lines between neighbouring sites. Of these tNx arereal bonds in double moleculea; while tN(z-x) are "non-bonds". Thenumber of their arrangements C is given by

In C = lN ~z In z - x In x - (z-x) In (z-x)~. (15)

But in N(I-x) points the condition must be fulfilled that the z joininglines are all non-bonds, and in each of the remaining points one bond andz-I non-bonds must meet. For one point the probability of fulfilment is~(z-x)jz~Z for the former condition and x~(z_x)jz~Z-l for the latter. Follow-ing Pauling we assume independence of the probabilities for the differentpoints. Taking into account the number of distributions of N(I-x) and Nxpoints, the fraction F of the arrangements C where the conditions arefulfilled is given by

Jn F = N O(I-x) z + x (z-I)t In)(z-x)jzH - (I-x) In (I-x)]. (16)

Addition of (15) and (16) then, rather surprisingly, yields (14).For x = 1 the similarity of the above method with Pauling's is complete.

For the A2B honey-comb lattice (z = 3, x = 1) Chang's approximation is8% low with respect to the result of section 8, and for the A3B squarelattice (z = 4, x = 1) about 10% with respect to the result of Fowler andRushbrooke 11).

REFERENCES

1) H. König, Z. Kristallogr. 105, 279-286, 1944.2) N. D. Lisgarten and M. Blackman, Nature, Lond. 178, 39-40, 1956.3) J. D. Bernal and R. H. Fowler, J. chem. Phys. 1, 515-548, 1933.

350 J. L. MEIJERING

4) L. Pauling, J. Ame», chem. Soc. 57, 2680-2684, 1935.ö) J. C. Slater, J. chem. Phys. 9, 16·33, 1941.8) N. Bjerrum, K. danske vidensk. SeIsk., Math-fys, Medd, 27, 1.56, 1951; Science

115, 385.390, 1952. '7) R. E. Rundle, J. chem. Phys. 21, 1311, 1953.8) W. N. Lipscomb. J. chem. Phys. 22, 344, 1954.9) H. Gränicher, Helv. phys. Acta 29, 212.215, 1956.10) K. S. Pitzer and J. Polissar, J. phys, Chem. 60, 1140·1142, 1956.11) R. H. Fowler and G. S. Rushbrooke, Trans, Faraday Soc. 33, 1272·1294, 1937.1lI) C. C. Stephenson and H. E. Adams, J. Amer. chem. Soc. 66, 1412·1416, 1944.13) W. F. Giauque and J. W. Stout, J. 'Amer, chem. Soc. 58, 1144·1150, 1936.14) E. A. Long and J. D. Kemp, J. Amer. chem, Soc. 58, 1829·1834, 1936.lij) P. G. Owston, J. Chim. phys. 50, C 13·18, 1953.18) M. Gordon and W. H. T. Davison, J. chem. Phys. 2~, 428·435, 1952.17) K. Lonsdale, Nature, Lond. 158, 582, 1946; R. F. Brill and S. Zaromb, Nature,

Lond. 173, 316·317, 1954.18) T. S. Chang, Proc. Camh, phil. Soc. 35, 265.292, 1939; Proc. Roy. Soc. A 169, 512.531,

1939.19) E .. A. Guggenheim, Mixtures, Clarendon Press, Oxford, 1952.20) Discussions to a paper by H. Gränicher, C. J accard, P. Scherrer and A. St eine-

mann, by F. C. Frank and by J. L. Meijering; Faraday Soc. Discussions: "TheMolecular Mechanism of Rate Processes in Solids", Amsterdam, April1957.